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Googology Wiki:LoG draft
This is a draft of a rewrite to the List of googologisms page, which will probably be renamed to List of large numbers. Not much here yet. ---- This article is a master list of all named or otherwise significant large numbers. Most googological numbers are defined by specialized notations, so here we group these numbers by notation, and the notations themselves are ordered according to their strength. It is therefore important to note that these orderings are rough approximations, and more importantly, these numbers are not necessarily in ascending order. With each number we list the author and the year the number was coined (if applicable). "n." is short for "named by," used when the number is defined by one person and named by another. Primitive recursive arithmetic These numbers are small enough to be succinctly described by primitive recursive notations. 0 to googol * Zero, 0 * Googolminex 10-10100 (Conway and Guy, 19??) * Champernowne constant C10 = 0.1234567891011... * One, 1 * Two, 2 * Three, 3 * Four, 4 * Five, 5 * Six, 6 * Seven, 7 * Eight, 8 * Nine, 9 * Ten, 10 * Hundred, 100 * Eleventy, 110 * Twelfty or long hundred, 120 * Gross, 144 * Baker's gross, 169 Googol to googolplex Peano arithmetic These numbers are small enough to be succinctly described by notations provably recursive in Peano arithmetic. * Little Graham (Ronald Graham, 1971; n. Sbiis Saibian 20??) * Graham's number (Ronald Graham, 1977) Hyper-E notation Extended Hyper-E notation BEAF: Linear arrays BEAF: Multidimensional arrays ATR0 These numbers are small enough to be succinctly described by notations provably recursive in arithmetical transfinite recursion. Cascading-E notation Larger computable numbers These numbers are too large to be described succinctly using a notation provably recursive in ATR0, but are still described by computable notations. Uncomputable numbers These numbers require hypercomputational systems to describe succinctly. Numbers here are in increasing order (heuristically decided based on strength of functions). * \(\Sigma(100)\) (busy beaver function, for comparison) * Fish number 4, \(\text{F}_4^{63}(3)\) ("Fish", 2002) * Xi function \(\Xi(10^6)\) (Adam Goucher, 20??) * Rayo's number \(\text{Rayo}(10^{100})\) (Agustín Rayo, 2007) * Fish number 7, \(\text{F}_7^{63}(10^{100})\) ("Fish", 2013) * BIG FOOT, \(\text{FOOT}^{10}(10^{100})\) ("Wojowu" and Nathan Ho, 2014; n. Sbiis Saibian) Transfinite numbers Ordinals and cardinals are two closely related systems extending the natural numbers to infinite values. Countable ordinals * \(\omega\) (Georg Cantor, 1883) * \(\omega^\omega\) ( of and recursive comprehension (RCA0)) * \(\varepsilon_0\) (Georg Cantor, 19?? ; PTO of Peano arithmetic and arithmetical comprehension (ACA0)) * \(\varepsilon_{\varepsilon_0}\) (PTO of ACA0 with the full second-order induction scheme (ACA)) * Cantor's ordinal \(\zeta_0\) (Georg Cantor, 19??; n. Sbiis Saibian 20??)]] * Feferman-Schütte ordinal \(\Gamma_0\) (n. Nik Weaver, 2005Weaver, Nik (2005), Predicativity beyond \(\Gamma_0\), ''http://arxiv.org/pdf/math/0509244v3.pdf ; PTO of arithmetical transfinite recursion (ATR0)) * Ackermann ordinal \(\vartheta(\Omega^2)\) (n. Nik Weaver, 2005) * Small Veblen ordinal \(\vartheta(\Omega^\omega)\) (Oswald Veblen, 1908 ; n. Nik Weaver, 2005 ; PTO of \(\text{ACA}_0+\Pi_2^1-\text{BI}\)) * Large Veblen ordinal \(\vartheta(\Omega^\Omega)\) (Oswald Veblen, 1908 ; n. Nik Weaver, 2005) * Bachmann-Howard ordinal \(\vartheta(\varepsilon_{\Omega + 1})\) (????, 19?? ; PTO of with axiom of infinity (KP)) * \(\psi_0(\Omega_\omega)\) (PTO of \(\Pi_1^1-\text{CA}_0\)) * Takeuti-Feferman-Buchholz ordinal \(\psi_0(\varepsilon_{\Omega_\omega + 1})\) (n. David Madore, 2008 ; PTO of ) * Ψ(ψᵢ(0)) (PTO of \(\Pi_1^1-\text{TR}_0\)) * \(\psi_{\Omega_1}(\varepsilon_{\text{I}+1})\) (PTO of KP + "there exists a recursively inaccessible cardinal" (KPI)) * \(\psi_{\Omega_1}(\varepsilon_{\text{M}+1})\) (PTO of KP + "there exists a recursively Mahlo cardinal" (KPM)) * \(\Psi_{\Omega_1}(0,\varepsilon_{\text{K}+1})\) (PTO of KP + \(\Pi\)-3 reflection) * Limit of Taranovsky's C function (conjectured but not proved equal to the PTO of Z2) * Proof-theoric ordinal of full second-order arithmetic (Z2) * Proof-theoric ordinal of Zermelo-Fraenkel set theory (ZF and ZFC) * Omega one chess \(\omega_1^{\mathfrak{Ch}}\) (C. D. A. Evans and Joel David Hamkins, 2013) ** Omega one chess with infinitely many pieces \(\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}\) ** Omega one of 3D chess \(\omega_1^{\mathfrak{Ch}_3}\) ** Omega one of 3D chess with infinitely many pieces \(\omega_1^ _3}\) * Church-Kleene ordinal \(\omega_1^{\text{CK}}\) (????, 19?? ; second and supremum of all possible proof-theoric ordinals) * Infinite time Turing machine ordinals (Joel David Hamkins and Andy Lewis, 1998) ** Supremum of all writable ordinals \(\lambda\) ** Supremum of all clockable ordinals \(\gamma\) ** Supremum of all eventually writable ordinals \(\zeta\) ** Supremum of all accidentally writable ordinals \(\Sigma\) Cardinals To represent cardinals as sets, most authors define a cardinal to be the least ordinal with that cardinality. Therefore \(\omega = \aleph_0\), \(\omega_1 = \aleph_1\), etc. and the choice of notation distinguishes cardinal arithmetic and ordinal arithmetic. All uncountable ordinals studied in the literature so far have been cardinals. * Aleph-zero, \(\aleph_0 = \omega\) * First uncountable ordinal \(\aleph_1 = \omega_1\) (sometimes \(\Omega\) is used) (Georg Cantor, 1883) * First omega fixed point \(\psi_\text{I}(0) = \sup(\Omega,\Omega_\Omega,\Omega_{\Omega_\Omega},...)\) (????, ????) * First inaccessible cardinal \(I\) (????, ????) * First Mahlo cardinal \(M\) (Paul Mahlo, 1911) * First weakly compact cardinal \(K\) (????, ????) * First indescribable cardinal (????, ????) * First rank-into-rank cardinal (????, ????) Miscellany These numbers that cannot be reasonably fit into the other categories for whatever reason. * ''q(5), Laver table pseudoinverse (Ralph Laver, 1992). Existence is an unsolved problem. Higher BEAF numbers BEAF beyond tetrational arrays is generally agreed to be ill-defined. If it were properly defined, these numbers would fall under "ATR0" and "Larger computable numbers." Sources